Improved upper and lower bounds for the point placement problem

The point placement problem is to determine the positions of a set of n distinct points, P = {p1, p2, p3, . . . , pn}, on a line uniquely, up to translation and reflection, from the fewest possible distance queries between pairs of points. Each distance query corresponds to an edge in a graph, called point placement graph (ppg), whose vertex set is P . The uniqueness requirement of the placement translates to line rigidity of the ppg. In this paper we show how to construct in 2 rounds a line rigid point placement graph of size 9n/7+O(1). This improves the result reported in [2] for 5-cycles. We also improve the lower bound on 2-round algorithms from 17n/16 [2] to 9n/8.